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Let $V$ be a vector space. We say that the inner product of vectors $\vec{v_1}$ and $\vec{v_2}$, written $<v_1,v_2>$, is a function that takes any two vectors to a real number and has the following additional properties:
1. (Symmetry or Conjugate Symmetry) When dealing with only real numbers, the following symmetry formula must hold: $$<x,y> = <y,x>,$$ and when dealing with complex numbers, the following conjugate symmetry formula holds: $$<x,y> = \overline{<y,x>},$$ where $\overline{<y,x>}$ denotes complex conjugation.

2. (Linearity in the first argument) The following formula holds for all scalars $\alpha,\beta$ and vectors $\vec{x},\vec{y},\vec{z}$: $$<\alpha \vec{x}+\beta \vec{y},\vec{z}> = \alpha<\vec{x},\vec{z}> + \beta<\vec{y},\vec{z}>.$$ Note in the case of real inner products we can factor out of the second term as $$<\vec{x},\alpha\vec{y}>=\alpha<\vec{x},\vec{y}>$$ but if we are dealing with a complex inner product, factoring out of the second term results in a conjugate factor: $$<\vec{x},\alpha \vec{y}> = \overline{\alpha}<\vec{x},\vec{y}>.$$ 3. (Positive definiteness) It is always true that $$< \vec{x},\vec{x} > \geq 0$$ and $<\vec{x},\vec{x}>=0$ if and only if $\vec{x}=\vec{0}$.

If $V$ is a vector space and $<\cdot,\cdot>$ is an inner product on $V$, then we say that $H=(V,<\cdot,\cdot>)$ is an inner product space.

Examples of inner product spaces
1. Let $\vec{x},\vec{y} \in \mathbb{R}^{n \times 1}$ with $\vec{x}=\begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}$ and $\vec{y}=\begin{bmatrix} y_1 \\ \vdots \\ y_n \end{bmatrix}$. Then the following formula defines an inner product: $$< \vec{x}, \vec{y} > = x_1y_1 + x_2y_2 + \ldots + x_ny_n = \displaystyle\sum_{k=1}^n x_ky_k.$$ 2. Let $\vec{p},\vec{q} \in \mathbb{P}$ (the vector space of polynomials). The following formula defines an inner product: $$<\vec{p},\vec{q}> = \displaystyle\int_0^{\infty} \vec{p}(x)\vec{q}(x)e^{-x} dx.$$ 5. Let $f,g \in C[0,1]$, the functions continuous on the interval $[0,1]$. The following formula defines an inner product: $$<f,g>=\displaystyle\int_0^1 f(x)g(x)dx.$$ 4. Let $\{a_k\}_{k=0}^{\infty}, \{b_k\}_{k=0}^{\infty} \in \ell^1(\mathbb{R})$, the set of sequences $\{x_k\}$ such that $\displaystyle\sum_{k=0}^{\infty} |x_k| < \infty$. The following formula defines an inner product: $$<\{a_k\},\{b_k\}> = \displaystyle\sum_{k=0}^{\infty} a_kb_k.$$ 5. Let $z_1, z_2$ be complex numbers with $z_1=a+bi$ and $z_2=c+di$ and $i^2=-1$. The following formula defines an inner product: $$<z_1,z_2> = z_1 \overline{z_2},$$ where $\overline{z_2}=c-di$ (called the complex conjugate of $z_2$).